The　security　of　many　fully　homomorphic　encryption　(FHE)　schemes　is　guaranteed　by　the　difficulty　of　the　approximate　greatest　common　divisor　(AGCD)　problem.　Therefore,　the　study　of　AGCD　problem　is　of　great　significance　to　the　security　of　the　fully　homomorphic　encryption.　This　paper　surveys　three　kinds　of　attacks　on　the　AGCD　problem,　i.e.　exhaustive　search　attack,　simultaneous　Diophantine　approximation　(SDA)　attack　and　the　orthogonal　lattice　(OL)　attack.　We　utilize　the　Number　Theory　Library　(NTL)　to　implement　the　SDA　attack　and　the　optimized　OL　attack　on　the　AGCD　problem.　Comparisons　are　performed　based　on　the　experimental　results　to　illustrate　that　the　exhaustive　search　attack　can　be　easily　defended　just　by　increasing　the　size　of　ρ.　And　increasing　the　length　of　the　public　key　is　the　most　effective　way　to　defend　SDA　attack　and　OL　attack.　Meanwhile,　we　concluded　that　the　success　rate　of　SDA　attack　and　OL　attack　can　be　improved　by　increasing　the　dimension　of　lattice　at　the　expense　of　a　certain　time　efficiency.　In　addition,　the　analysis　and　experiments　show　that　the　fully　homomorphic　computing　efficiency　of　FHE　scheme　can’t　be　improved　by　simply　increasing　the　private　key　without　appropriately　increasing　the　size　of　public　key.　Otherwise,　the　FHE　scheme　is　vulnerable　to　OL　and　SDA　attack.　Besides,　experimental　results　show　that　optimized　OL　attack　performs　better　than　both　classical　OL　attack　and　SDA　attack　in　terms　of　attack　success　rate　and　the　time　efficiency.　©　2019,　Springer　Nature　Singapore　Pte　Ltd.